Inverse eigenvalue problem for a class of Dirac operators with discontinuous coefficient
© Mamedov and Akcay; licensee Springer. 2014
Received: 30 November 2013
Accepted: 25 April 2014
Published: 13 May 2014
In this paper, the inverse problem of recovering the coefficient of a Dirac operator is studied from the sequences of eigenvalues and normalizing numbers. The theorem on the necessary and sufficient conditions for the solvability of this inverse problem is proved and a solution algorithm of the inverse problem is given.
The inverse problem for the Dirac operator with separable boundary conditions was completely solved by two spectra in [1, 2]. The reconstruction of the potential from one spectrum and norming constants was investigated in . For the Dirac operator, the inverse periodic and antiperiodic boundary value problems were given in [4–6]. Using the Weyl-Titschmarsh function, the direct and inverse problems for a Dirac type-system were developed in [7, 8]. Uniqueness of the inverse problem for the Dirac operator with a discontinuous coefficient by the Weyl function was studied in  and discontinuity conditions inside an interval were worked out in [10, 11]. The inverse problem for weighted Dirac equations was obtained in . The reconstruction of the potential by the spectral function was given in . For the Dirac operator with peculiarity, the inverse problem was found in . Inverse nodal problems for the Dirac operator were examined in [15, 16]. In the case of potentials that belong entrywise to , for some , the inverse spectral problem for the Dirac operator was studied in , and in this work, not only the Gelfand-Levitan-Marchenko method but also the Krein method  was used. In the positive half line, the inverse scattering problem for the Dirac operator with discontinuous coefficient was analyzed in . Besides, in a finite interval, for Sturm-Liouville operator inverse problem has widely been developed (see [20–22]). The inverse problem of the Sturm-Liouville operator with discontinuous coefficient was worked out in [23, 24] and discontinuous conditions inside an interval were obtained in . In the mathematical and physical literature, the direct and inverse problems for the Dirac operator are widespread, so there are numerous investigations as regards the Dirac operator. Therefore, we can mention the studies concerned with a discontinuity, which is close to our topic, in the references list.
In this paper, our aim is to solve the inverse problem for the Dirac operator with a piecewise continuous coefficient on a finite interval. Let and () be, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2). The quantities () are called spectral data. We can state the inverse problem for a system of Dirac equations in the following way: knowing the spectral data () to indicate a method of determining the potential and to find necessary and sufficient conditions for () to be the spectral data of a problem (1), (2). In this paper, this problem is completely solved.
We give a brief account of the contents of this paper in the following section.
The following two theorems are obtained by Huseynov and Latifova in .
- (ii)The eigen vector-functions of problem (1), (2) can be represented in the form
- (iii)The normalizing numbers of problem (1), (2) have the form(9)
Theorem 2 (i) The system of eigen vector-functions () of problem (1), (2) is complete in space .
moreover, the series converges uniformly with respect to .
where () are zeros of the function and δ is a sufficiently small number.
is found. We demonstrate by using Lemma 6, Lemma 9, and Lemma 10 that () are spectral data of the boundary value problem (1), (2). Then necessary and sufficient conditions for the solvability of problem (1), (2) are obtained in Theorem 11. Finally, we give an algorithm of the construction of the function by the spectral data ().
Note that throughout this paper, denotes the transposed matrix of ϕ.
3 Fundamental equation
where and are, respectively, eigenvalues and normalizing numbers of the boundary value problem (1), (2) when .
is obtained. □
Lemma 4 For each fixed , (13) has a unique solution .
Now, we shall prove that is invertible, i.e. has a bounded inverse in .
Since the system () is complete in , we have , i.e. . For invertible in , is obtained. □
Proof According to (14) and (15), and . Then, from the fundamental equation (13), we have . It follows from (4) that a.e. on . □
5 Reconstruction by spectral data
Let the real numbers () of the form (8) and (9) be given. Using these numbers, we construct the functions and by (14) and (15) and determine from the fundamental equation (13).
Now, let us construct the function by (3) and the function by (4). From , and have a derivative in both variables and these derivatives belong to .
is obtained. For , from (3) we get (30). □
is obtained, i.e., (47) is valid. □
The system is minimal in and consequently by (3), the system is minimal in . Hence and we obtain (56). □
we find , and then is obtained. □
Theorem 11 For the sequences () to be the spectral data for a certain boundary value problem of the form (1), (2) with , it is necessary and sufficient that the relations (8) and (9) hold.
Proof Necessity of the problem is proved in . Let us prove the sufficiency. Let the real numbers () of the form (8) and (9) be given. It follows from Lemma 6, Lemma 9, and Lemma 10 that the numbers () are spectral data for the constructed boundary value problem . The theorem is proved. □
By the given numbers () the functions and are constructed, respectively, by (14) and (15).
The function is found from (13).
is calculated by (4).
This work is supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK).
- Gasymov MG, Levitan BM: The inverse problem for the Dirac system. Dokl. Akad. Nauk SSSR 1966, 167: 967-970.MathSciNetGoogle Scholar
- Gasymov MG, Dzabiev TT: Solution of the inverse problem by two spectra for the Dirac equation on a finite interval. Dokl. Akad. Nauk Azerb. SSR 1966, 22(7):3-6.MathSciNetGoogle Scholar
- Dzabiev TT: The inverse problem for the Dirac equation with a singularity. Dokl. Akad. Nauk Azerb. SSR 1966, 22(11):8-12.MathSciNetGoogle Scholar
- Misyura TV: Characteristics of spectrums of periodical and antiperiodical boundary value problems generated by Dirac operation. 31. In II. Teoriya funktsiy, funk. analiz i ikh prilozheiniya. Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine, Kharkov; 1979:102-109.Google Scholar
- Nabiev IM: Solution of a class of inverse problems for the Dirac operator. Trans. Natl. Acad. Sci. Azerb. 2001, 21(1):146-157.MathSciNetGoogle Scholar
- Nabiev IM: Characteristic of spectral data of Dirac operators. Trans. Natl. Acad. Sci. Azerb. 2004, 24(7):161-166.MathSciNetGoogle Scholar
- Sakhnovich A: Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems. Inverse Probl. 2006, 22(6):2083-2101. 10.1088/0266-5611/22/6/011MathSciNetView ArticleGoogle Scholar
- Fritzsche B, Kirstein B, Roitberg IY, Sakhnovich A: Skew-self-adjoint Dirac system with a rectangular matrix potential: Weyl theory, direct and inverse problems. Integral Equ. Oper. Theory 2012, 74(2):163-187. 10.1007/s00020-012-1997-1MathSciNetView ArticleGoogle Scholar
- Latifova AR: The inverse problem of one class of Dirac operators with discontinuous coefficients by the Weyl function. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 2005, 22(30):65-70.MathSciNetGoogle Scholar
- Amirov RK: On system of Dirac differential equations with discontinuity conditions inside an interval. Ukr. Math. J. 2005, 57(5):712-727. 10.1007/s11253-005-0222-7MathSciNetView ArticleGoogle Scholar
- Huseynov HM, Latifova AR: The main equation for the system of Dirac equation with discontinuity conditions interior to interval. Trans. Natl. Acad. Sci. Azerb. 2008, 28(1):63-76.MathSciNetGoogle Scholar
- Watson BA: Inverse spectral problems for weighted Dirac systems. Inverse Probl. 1999, 15(3):793-805. 10.1088/0266-5611/15/3/311View ArticleGoogle Scholar
- Mamedov SG: The inverse boundary value problem on a finite interval for Dirac’s system of equations. Azerb. Gos. Univ. Ucen. Zap. Ser. Fiz-Mat. Nauk 1975, 5: 61-67.Google Scholar
- Panakhov ES: Some aspects inverse problem for Dirac operator with peculiarity. Trans. Natl. Acad. Sci. Azerb. 1995, 3: 39-44.Google Scholar
- Yang CF, Huang ZY: Reconstruction of the Dirac operator from nodal data. Integral Equ. Oper. Theory 2010, 66: 539-551. 10.1007/s00020-010-1763-1View ArticleGoogle Scholar
- Yang CF, Pivovarchik VN: Inverse nodal problem for Dirac system with spectral parameter in boundary conditions. Complex Anal. Oper. Theory 2013, 7: 1211-1230. 10.1007/s11785-011-0202-xMathSciNetView ArticleGoogle Scholar
- Albeverio S, Hryniv R, Mykytyuk Y: Inverse spectral problems for Dirac operators with summable potentials. Russ. J. Math. Phys. 2005, 12(14):406-423.MathSciNetGoogle Scholar
- Krein MG: On integral equations generating differential equations of the second order. Dokl. Akad. Nauk SSSR 1954, 97: 21-24.MathSciNetGoogle Scholar
- Mamedov KR, Çöl A: On an inverse scattering problem for a class Dirac operator with discontinuous coefficient and nonlinear dependence on the spectral parameter in the boundary condition. Math. Methods Appl. Sci. 2012, 35(14):1712-1720. 10.1002/mma.2553MathSciNetView ArticleGoogle Scholar
- Marchenko VA: Sturm-Liouville Operators and Applications. Am. Math. Soc., Providence; 2011.Google Scholar
- Freiling G, Yurko V: Inverse Sturm-Liouville Problems and Their Applications. Nova Science Publishers, New York; 2008.Google Scholar
- Guliyev NJ: Inverse eigenvalue problems for Sturm-Liouville equations with spectral parameter linearly contained in one of the boundary conditions. Inverse Probl. 2005, 21: 1315-1330. 10.1088/0266-5611/21/4/008MathSciNetView ArticleGoogle Scholar
- Mamedov KR, Cetinkaya FA: Inverse problem for a class of Sturm-Liouville operator with spectral parameter in boundary condition. Bound. Value Probl. 2013. 10.1186/1687-2770-2013-183Google Scholar
- Akhmedova EN, Huseynov HM: On solution of the inverse Sturm-Liouville problem with discontinuous coefficients. Trans. Natl. Acad. Sci. Azerb. 2007, 27(7):33-44.MathSciNetGoogle Scholar
- Yang CF, Yang XP: An interior inverse problem for the Sturm Liouville operator with discontinuous conditions. Appl. Math. Lett. 2009, 22: 1315-1319. 10.1016/j.aml.2008.12.001MathSciNetView ArticleGoogle Scholar
- Latifova AR: On the representation of solution with initial conditions for Dirac equations system with discontinuous coefficients. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 2002, 16(24):64-68.MathSciNetGoogle Scholar
- Huseynov HM, Latifova AR: On eigenvalues and eigenfunctions of one class of Dirac operators with discontinuous coefficients. Trans. Natl. Acad. Sci. Azerb. 2004, 24(1):103-112.MathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.